In standard collinear perturbative QCD (pQCD) factorization framework, hadronization of charged particles is described by collinear fragmentation functions (FFs), denoted as \(D_{c}^{h}(z, \mu )\) , where \(z\)  is the momentum fraction of an outgoing parton \(c\)  carried by a hadron \(h\)  (see [1]} for a review). In this framework, single inclusive hadron production in \(p\)\(p\)  collisions factorizes into the the short distance hard scattering of partons and the long distance dynamics described by collinear FFs and collinear parton distribution functions (PDFs). The latter, denoted by \(f_{a/p} (x, \mu )\) , parameterize the momentum fraction \(x\)  of an incoming proton carried by a parton \(a\)  taking a part in hard scattering processes. The FFs and PDFs are non-perturbative (NP) objects that can only be measured in experiments, in contrast to the single inclusive jet production that is perturbatively calculable in pQCD. Jet fragmentation functions (JFFs) are experimental observables that measure the longitudinal momentum fraction of a jet carried by hadrons inside the jets. Within the SCET framework, the JFFs are defined in such a way that they can probe the collinear FFs. Analogously, TMD JFFs can access standard definition of TMD FFs measured in e+e- and SIDIS [2]}. Theoretical development of this kind is crucial in the following sense. TMD factorization breaking is predicted for jet production in p+p [3]}, in which case non-perturbative TMD distribution and fragmentation functions can no longer be identified as universal objects in different collision processes. As such, interpretation of these measurements has to be carried out with caution and test of factorization breaking and universality of non-perturbative FFs across different collision systems are indispensible.
LHCb data provide constraints on these NP physics objects in kinematic regimes complementary to those accessible at central pseudorapidities, e.g. a region of \(x\)  smaller than \(10^{-4}\)  for TMD PDFs. The Drell Yan (DY) measurements with the \(Z \rightarrow \mu \mu \)  channel [4]}, [5]}, [6]} at LHCb have been used in a recent global analysis [7]} of TMDs. The DY angular distributions data affected by Boer-Mulders TMD PDF are also expected to constrain TMD PDFs and test their universality.
